Abstract

This study focuses on the potential improvement of environmental variables modelling by using linear state-space models, as an improvement of the linear regression model, and by incorporating a constructed hydro-meteorological covariate. The Kalman filter predictors allow to obtain accurate predictions of calibration factors for both seasonal and hydro-meteorological components. This methodology can be used to analyze the water quality behaviour by minimizing the effect of the hydrological conditions. This idea is illustrated based on a rather extended data set relative to the River Ave basin (Portugal) that consists mainly of monthly measurements of dissolved oxygen concentration in a network of water quality monitoring sites. The hydro-meteorological factor is constructed for each monitoring site based on monthly precipitation estimates obtained by means of a rain gauge network associated with stochastic interpolation (kriging). A linear state-space model is fitted for each homogeneous group (obtained by clustering techniques) of water monitoring sites. The adjustment of linear state-space models is performed by using distribution-free estimators developed in a separate section.

Keywords

Notes

Acknowledgements

The authors would like to thank to Eng. Pimenta Machado from the Portuguese Regional Directory for the Northern Environment and Natural Resources, and to Eng. Cláudia Brandão from the Portuguese Institute of Water, for sharing their skills and experiences and for supplying the monitored data. A. Manuela Gonçalves acknowledges the financial support provided by the Research Centre of Mathematics of the University of Minho through the FCT Pluriannual Funding Program.

Appendix

Distribution-free estimators for the mean and for the transition matrix

In the parameters estimation of state-space models were performed distribution-free estimators developed from the original work by Costa and Alpuim (2010). However, in that work it was proposed a distribution-free estimator for state-space models with univariate observations. Thus, a straightforward generalization of these estimators is presented in order to allow their application to a class of multivariate state-space models that largely covers the present work’s needs.

it is assumed a set of observations \({{\mathcal Y}_n=(\mathbf{Y}_1, {\mathbf Y}_2, \ldots, {\mathbf Y}_n),}\) and regular matrices of known constants \({\mathbf H}_1, {\mathbf H}_2, \ldots, {\mathbf H}_n\) are available. The mean vector \({\mathbf \mu}\) can be easily estimated by the method of moments, i.e., \(\widehat{\mathbf \mu}=n^{-1}\sum_{t=1}^n {\mathbf H}_t^{-1}{\mathbf Y}_t.\)

As variables Yt are not stationary, we are not under the usual conditions of the consistency of generalized method of moments. Thus, it is necessary to establish additional conditions to guarantee this consistency. By construction, the estimator \(\widehat{\mathbf \mu}\) of the mean vector is unbiased, so we can guarantee its consistency by proving that \(var(\widehat{\mathbf \mu})\rightarrow {\mathbf 0}\) when \(n\rightarrow+\infty,\) and thus establishing sufficient conditions. Covariance matrix of \(\widehat{\mathbf \mu}\) is given by

it is sufficient to admit the additional condition |ht,(i,j)−1| < c for all \(t=1,2,.., i,j=1,2,\ldots,m\) and for some positive constant c, where ht,(i,j)−1 represents the (i, j) element of Ht−1 matrix.

The autoregressive matrix \(\mathbf{\Upphi}\) is estimated by means of covariance structure of process \({\mathbf H}_t^{-1}{\mathbf Y}_t.\) We see that

In a VAR(1) process, the relation \(\mathbf{\Upgamma}_k=\mathbf{\Upphi \Upgamma}_{k-1}\) is valid, for k = 1, 2, .... Thus, we proposed the autoregressive matrix estimator \(\widehat{\Upphi}\) based on the least squares method of these equations by taking \(k=1,2,\ldots,\ell_\mathbf{\Upphi}.\) Thus, we have

By construction, the autoregressive matrix estimator is consistent, since \(\widehat{\mathbf{\Upgamma}}_k\) is a consistent estimator of \(\mathbf{\Upgamma}_k.\) Whereas we have proposed a consistent estimator to \(\mathbf{\mu},\) we consider that the mean vector \(\mathbf{\mu}\) is known. To analyse the consistency of \(\widehat{\mathbf{\Upgamma}}_k\) we have

Under the previously established condition, the last three parcels converge in probability to a null matrix. Indeed, by defining the second parcel as \({\mathbf A}=[A_{ij}]_{i,j=1,2,\ldots,m}\) and, with some algebraic manipulation, we have

If the additional condition |ht,(i,j)−1| < c is valid, this parcel tends to 0 when \(n\rightarrow+\infty. \) In a similar way, we defined the third parcel by \(\mathbf{B}=[B_{ij}]_{i,j=1,2,\ldots,m}\) with elements given by

Again, we guarantee that Bij = Op through the same condition |ht,(i,j)−1| < c. As we shall see, this condition is a sufficient condition, as the last parcel also tends to a null matrix. Indeed, if we denote the last parcel as \(\mathbf{C}=[C_{ij}]_{i,j=1,2,\ldots,m}, \) we have

These results allow us to conclude that if |ht,(i,j)−1| < c, the estimator \(\widehat{\mathbf{\Upgamma}}_k\) is consistent to \(\mathbf{\Upgamma},\) when we replace the mean vector \(\mathbf{\mu}\) by a consistent estimator.

Distribution-free estimators to noise variances

The estimation of covariance matrices of errors terms et and εt is an important and difficult step at the same time. At times, the recursive procedures applied to the obtained Gaussian likelihood estimates diverge or produce non-positive semidefined matrices. Sometimes, these problems occur when the initial solution is not as close to estimates as necessary. We propose an estimator to \(\mathbf{\Upsigma}_\mathbf{\varepsilon}\) based on covariance structure of a VAR(1) stationary process.

We know that the relation \(\mathbf{\Upsigma}_{\mathbf{\beta}}=\mathbf{\Upphi\Upsigma\Upphi}^{\prime}+ \mathbf{\Upsigma}_{\mathbf{\varepsilon}}\) is valid in a VAR(1) stationary process, or by applying the Kronecker product ⊗ and the operator vec

The consistency of \(\widehat{\mathbf \Upsigma}_{\mathbf \varepsilon}\) is guaranteed under the same conditions of the consistency of \(\widehat{\mathbf \Updelta}_k.\) As we have seen, a sufficient condition for this is |ht,(i,j)−1| < c.

In order to estimate the covariance matrix \({\mathbf \Upsigma}_{\mathbf e},\) we defined

As the matrix \({\mathbf \Upsigma}_{\mathbf e}\) is symmetric, it is necessary to adopt the same procedure as in the estimation of \({\mathbf \Upsigma}_{\mathbf \varepsilon}.\) Thus, we estimated the m + m(m − 1)/2 elements of the covariance matrix.

If we have a consistent estimator to \({\mathbf \Upsigma}_{\mathbf \beta},\) for example given by the proposed estimators to \({\mathbf \Upphi}\) and \({\mathbf \Upsigma}_{\mathbf \varepsilon},\) the consistency of \(\widehat{\mathbf \Upsigma}_{\mathbf e}\) boils down to the limit of variance of each element of \(vec({\mathbf \Upupsilon})= n vec(\widehat{\mathbf \Uppsi})^{\prime}[\sum_{t=1}^n({\mathbf H}_t^{-1}\otimes {\mathbf H}_t^{-1})^{\prime}]^{-1}.\) The variance of the (i, j) element of \({\mathbf \Upupsilon}\) is given by

where ht,(i,j)−1 represents the (i,j) element of the matrix \({\mathbf H}_t^{-1}\) and aij the (i, j) element of the matrix \([\sum_{t=1}^n({\mathbf H}_t^{-1}\otimes {\mathbf H}_t^{-1})^{\prime}]^{-1}.\)

For simplicity, we adopt βt,i − μi = βt,i*. If we take in account that the states \({\mathbf \beta}_t\) are uncorrelated to noise \({\mathbf e}_s\) for all t and s, the previous expression can be decomposed into four parcels. The first parcel has the form

So, if we admit that the elements of matrix \(\mathbf{H}^{-1}_t\) are limited as c1 < |ht,(i,j)−1| < c2, where c1 and c2 are positive constants, it follows that this term is an Op. In addition to these conditions on ht,(i,j)−1, if we ensure that the vector of error et is stationary of fourth-order, then we conclude that the last parcel of variance of the (i, j) element of \({\mathbf \Upupsilon}\) is an Op, too.

Thus, under the additional stationarity conditions of fourth-order on the vector of disturbances and the above restrictions on the elements of the matrices \({\mathbf H}_t^{-1}, \) the proposed distribution-free estimator to \({\mathbf \Upsigma}_{\mathbf e}\) is consistent.

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